The theory and practice of topological data analysis (TDA) over the past decade has been dominated by the notion and application of persistent homology. Given a representation of the persistent homology of a filtered structure, it is easy to compute the Euler characteristic curve (ECC) which gives the value of the Euler characteristic of the structure at each point of the filtration. Although ECCs contains less information than, for example, persistence diagrams, they are typically much easier to compute numerically, using techniques not much different to those used by Euler almost three centuries ago. From a practical point of view, ECCs often seem to contain most of the useful information that is necessary for data analysis and, from a more theoretical point of view, it has turned out to be much easier to study ECCs for random structures than persistent homology or even Betti numbers at some point of a filtration. In the early 1990's, more or less a decade before the appearance of modern TDA, researchers in medical imaging coined the term "Topological Inference" to describe a collection of techniques that exploited the ease of computation and availability of theoretical results for ECCs that were used to analyse PET, MRI and fMRI data. In this talk I will describe the relation between persistent homology and ECCs in a number of practical and theoretical scenarios, as well as the somewhat different but often quite parallel approaches of TDA and Topological Inference.